Integrate over Regions

Integrate over basic geometric regions.

 In[1]:= X\[ScriptCapitalR] = Polygon[{{0, 0}, {1, 2}, {4, 0}}];
 In[2]:= XIntegrate[Sin[x] Sin[y], {x, y} \[Element] \[ScriptCapitalR]]
 Out[2]=

Visualize the region.

 In[3]:= XPlot3D[Sin[x] Sin[y], {x, y} \[Element] \[ScriptCapitalR]]
 Out[3]=

Integrate over regions in higher dimensions.

 In[4]:= XIntegrate[1, {x[1], x[2], x[3], x[4], x[5]} \[Element] Ball[{0, 0, 0, 0, 0}, r]]
 Out[4]=
 In[5]:= XNIntegrate[1, {x[1], x[2], x[3], x[4], x[5]} \[Element] Ball[{0, 0, 0, 0, 0}, 1]]
 Out[5]=

Integrate over implicit regions.

 In[6]:= X\[ScriptCapitalR] = ImplicitRegion[x y z <= 1, {{x, -5, 5}, {y, -5, 5}, {z, -5, 5}}];
 In[7]:= XIntegrate[1, {x, y, z} \[Element] \[ScriptCapitalR]]
 Out[7]=
 Out[8]=

Integrate over parametric regions.

 In[9]:= X\[ScriptCapitalR] = ParametricRegion[{Sin[u], Cos[u], u/10}, {{u, 0, 20}}];
 In[10]:= XIntegrate[1, {x, y, z} \[Element] \[ScriptCapitalR]]
 Out[10]=
 Out[11]=

Mathematica

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